{"paper":{"title":"Convergence of the solutions of the discounted equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Albert Fathi, Andrea Davini, Maxime Zavidovique, Renato Iturriaga","submitted_at":"2014-08-28T13:38:34Z","abstract_excerpt":"We consider a continuous coercive Hamiltonian $H$ on the cotangent bundle of the compact connected manifold $M$ which is convex in the momentum. If $u_\\lambda:M\\to\\mathbb R$ is the viscosity solution of the discounted equation $$\n  \\lambda u_\\lambda(x)+H(x,d_x u_\\lambda)=c(H), $$ where $c(H)$ is the critical value, we prove that $u_\\lambda$ converges uniformly, as $\\lambda\\to 0$, to a specific solution $u_0:M\\to\\mathbb R$ of the critical equation $$ H(x,d_x u)=c(H). $$ We characterize $u_0$ in terms of Peierls barrier and projected Mather measures."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.6712","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}